Question

A square and an equilateral triangle have equal perimeters. If a diagonal of the square is
24root2 cm, then find the area of the triangle. ​

Answer 1

Given :

• Diagonal of the square = 24√2 cm

• Perimeter of the square = Perimeter of the Equilateral triangle

To find :

• The area of the triangle.

Solution :

To find the area of the triangle , first we have to find the side of the triangle.

From the information that Perimeter of the triangle is equal to the perimeter of the square.

So, if we can find the perimeter of the square we can easily find the side the triangle .

But first let us find the side of the square.

To find the side of the square :-

Using the formula for Diagonal of a square and substituting the values in it , we get :

Let the side of the square be a cm.

$\boxed{:\implies \bf{Diagonal = \sqrt{2} \times side}}$

$:\implies \bf{24\sqrt{2} = \sqrt{2} \times a}$

$:\implies \bf{\dfrac{24\sqrt{2}}{\sqrt{2}} = a}$

$:\implies \bf{\dfrac{24\sqrt{\not{2}}}{\sqrt{\not{2}}} = a}$

$:\implies \bf{24 = a}$

$\underline{\therefore \bf{a = 24\:cm}}$

Hence, the side of the square is 24 cm.

Now , to find the Perimeter of the square :-

Using the formula for perimeter of a square and substituting the values in it,we get :-

$\boxed{:\implies \bf{P = 4 \times side}}$

$:\implies \bf{P = 4 \times 24}$

$:\implies \bf{P = 96}$

$\underline{\therefore \bf{P = 96\:cm}}$

Hence, the Perimeter of the square is 96 cm.

To find the side of the equilateral triangle :-

We know that the perimeter of the square is equal to the Perimeter of the triangle

Hence, using the formula for perimeter of a triangle and substituting the values in it, we get :

Let the side of the triangle be x.

$\boxed{:\implies \bf{P = 3 \times side}}$

$:\implies \bf{96 = 3 \times x}$

$:\implies \bf{\dfrac{96}{3} = x}$

$:\implies \bf{32 = x}$

$\underline{\therefore \bf{a = 32\:cm}}$

Hence, the side of the triangle is 32 cm.

Now,

To find the area of the triangle :-

Using the formula for area of an equilateral triangle and substituting the values in it,we get :

$\boxed{:\implies \bf{A = \dfrac{\sqrt{3} \times (side)^{2}}{4}}}$

$:\implies \bf{A = \dfrac{\sqrt{3} \times 32^{2}}{4}}$

$:\implies \bf{A = \dfrac{\sqrt{3} \times 1024}{4}}$

$:\implies \bf{A = \dfrac{1024\sqrt{3}}{4}}$

$:\implies \bf{A = 256\sqrt{3}}$

$\underline{\therefore \bf{A = 256\sqrt{3}\:cm^{2}}}$

Hence, the area of the triangle is 256√3 cm².

Answer 2

$\sf{ \purple{\Large{Given:}}}$

◔ Perimeter of a square = Perimeter of an equilateral triangle.

◔ Diagonal of the square = 24√2 cm

$\sf{ \purple{\Large{To \: Find:}}}$

๑ The area of the triangle.

$\sf{ \purple{\Large{Solution:}}}$

To find the area of equilateral triangle first we will find it's side.

Let the square has side "a" and triangle as side "t".

According to the question,

We know perimeter of square is equal to perimeter of an equilateral triangle, i.e.,

Formula for perimeter of square

$\underline{ \boxed{ \sf{Perimeter = 4 \times side}}}$

Formula for perimeter of equilateral triangle

$\underline{ \boxed{ \sf{Perimeter = 3 \times side}}}$

Therefore, we have

4a = 3t

$\sf{ \dfrac{4a }{3} } = t$

$\sf{t = \dfrac{4}{3} \times a}$

Therefore, for finding the "t" i.e., the side of equilateral triangle, we must know "a" i.e., the side of square.

For finding the side of square, we will take its diagonal and then find....

Formula for calculating the diagonal of square.

$\underline{ \boxed{ \sf{Diagonal \: of \: square = \sqrt{2} \times side}}}$

We already know the diagonal of square as per the question, so we have:

$\sf24 \sqrt{2} = \sqrt{2} \times a$

$\sf \dfrac{24 \sqrt{2} }{ \sqrt{2} } = a$

$\sf{a = 24}$

So,

$\sf{a = 24 \: cm}$

Now we can find the value of "t" i.e., the side of an equilateral triangle.

$\sf{t = \dfrac{4}{3} \times a \: \: (from \: above)}$

$\sf{t = \dfrac{4}{3} \times 24}$

$\sf{t = 32}$

So,

$\sf{t = 32 \: cm}$

Now, for we will use formula for finding area of an equilateral triangle.

Formula for calculating area of an equilateral triangle

$\underline{ \boxed{ \sf{Area \: of \: equilateral \: triangle \: = \dfrac{ \sqrt{3} }{4} \times {(side)}^{2} }}}$

$\sf{Area= \dfrac{ \sqrt{3} }{4} \: {t}^{2} }$

Substituting the values, we have:

$\sf{ = \dfrac{ \sqrt{3} }{4} \times {(32)}^{2} }$

$\sf = \dfrac{ \sqrt{3} }{4} \times1024$

$\sf = 256 \sqrt{3}$

$\sf = 443.405 \: cm$

∴ Area of an equilateral triangle =

$\green{\underline{ \boxed{ \sf{256 \sqrt{3} }}}}$

Or,

$\green{\underline{ \boxed{ \sf{443.405 \: cm }}}}$

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